Staff: Mentor

They're ways of quantifying how different the orbit is from a circular orbit. The physical significance comes from the fact that an elliptical orbit is physically different from a circular orbit (the distance from the center changes with time, the speed of the orbiting body changes with time).

What are the physical significances of the eccentricity and of the semi-latus rectum of the orbital ellipse?

if you look up the shape of planetary orbits the eccentricity significantly describe the nature of path and the eccentricity alongwith semi latus rectum are related to the distance of minimum approach or maximum distance for say an elliptical path and in turn it gets related to time period of the planet.

Assuming a simple harmonic oscillation of the gravitational potential centered at -GM/l, and with extrema labeled 1 and 2, then:

a. the shifts in the potential are equal and opposite:

-(GM/l - GM/r1) = -(-(GM/l - GM/r2))

Dividing by GM reveals l as the harmonic mean of r1 and r2.

b. Dividing the equation above by GM/l , we get:

1/r1 -1 = 1 - 1/r2 = e

This is the magnitude of the fractional shifts of the gravitational potential. It is also the eccentricity.

c. The amplitude of the oscillation is eGM/l . So the potential at a distance r may be expressed as:

- GM/r = - (GM+eGMcosq)/l

were q is a state variable of the oscillation. This equation may be rewritten as:

r = l/(1 + ecosq)

The couple (q, r) alternates between the extrema (0, r1) and (π, r2). These are 'collinear' with the 'origin'. So assigning the quantity 2A to the 'length' between these points, we get:

2A = l/(1 + e) + l/(1 - e)

and l = A(1 - e2)

So we may express the orbital radius as:

r = A(1 - e2)/(1 + ecosq)

This is, in polar coordinates, the equation of an ellipse. For the orbital ellipse, q is the true anomaly. Also l is the semi-latus rectum and is shown here to be the orbital radius at the center of the simple harmonic oscillation of the gravitational potential.